Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers

TL;DR

This work proves that the vector of first-order Riesz transforms $oldsymbol{ R}=ig(X_joldsymbol{Δ}^{-1/2}ig)_{j=0}^d$ on a Damek–Ricci space $S$ is weak type $(1,1)$ and bounded on all $L^p(S)$ for $p eq1$, unifying low-$p$ and high-$p$ regimes. The authors derive sharp heat-kernel estimates and kernel asymptotics, and they introduce an operator-valued joint functional calculus for commuting self-adjoint operators to handle the challenging $p o(2, fty)$ range. A key conceptual advance is the operator-valued spectral multiplier theorem, which reduces the $L^p$-boundedness of convolution by infinity-terms to bounds on operator-valued symbols $M_0,M_{rak v},M_{rak z}$ for the joint calculus of the sub-Laplacian on $N$ and central derivatives. This framework depends on a precise Gelfand-transform analysis on $N$, a detailed analysis of the radial structure, and a careful decomposition into local and infinity parts, enabling boundedness results on non-doubling spaces with exponential volume growth and offering tools potentially applicable to broader families of Lie groups and subelliptic operators.

Abstract

Let $Δ= \nabla^* \nabla$ be the distinguished Laplacian on a Damek-Ricci space. We prove the $L^{p}$-boundedness of the vector of first-order Riesz transforms $\nabla Δ^{-1/2}$ in the full range $p\in(1,\infty)$. The most demanding part of the proof is establishing the boundedness for $p \in (2,\infty)$; this is obtained as a consequence of an operator-valued spectral multiplier theorem for the joint functional calculus of a commuting system of self-adjoint operators, which we prove here and may be of independent interest.

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2